Derivative of e^-t

We now understand the derivative of e^-t. It is commonly represented as d/dt (e^-t) or (e^-t)', and its value is -e^-t. The function e^-t has a clearly defined derivative, indicating it is differentiable over the real numbers. The key concepts are mentioned below: Exponential Function: e^-t represents an exponential decay function. Chain Rule: A rule for differentiating composite functions like e^-t. Negative Sign: The negative exponent indicates that the function decreases as t increases.

The derivative of e^-t can be denoted as d/dt (e^-t) or (e^-t)'. The formula we use to differentiate e^-t is: d/dt (e^-t) = -e^-t The formula applies to all t in the real number set.

We can derive the derivative of e^-t using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: By First Principle Using Chain Rule We will now demonstrate that the differentiation of e^-t results in -e^-t using the above-mentioned methods: By First Principle The derivative of e^-t can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of e^-t using the first principle, we will consider f(t) = e^-t. Its derivative can be expressed as the following limit. f'(t) = limₕ→₀ [f(t + h) - f(t)] / h … (1) Given that f(t) = e^-t, we write f(t + h) = e^-(t + h). Substituting these into equation (1), f'(t) = limₕ→₀ [e^-(t + h) - e^-t] / h = limₕ→₀ [e^-t (e^-h - 1)] / h = e^-t limₕ→₀ [(e^-h - 1) / h] Using limit properties, the limit simplifies to -1. f'(t) = -e^-t. Hence, proved. Using Chain Rule To prove the differentiation of e^-t using the chain rule, We use the formula: u = -t so that e^-t = e^u By the chain rule, the derivative of e^u with respect to t is (de^u/du) * (du/dt). We know that de^u/du = e^u and du/dt = -1. Substituting, we find: d/dt (e^-t) = e^(-t) * (-1) = -e^-t

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like e^-t. For the first derivative of a function, we write f′(t), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(t). Similarly, the third derivative, f′′′(t), is the result of the second derivative, and this pattern continues. For the nth derivative of e^-t, we generally use fⁿ(t) for the nth derivative of a function f(t), which tells us the change in the rate of change (continuing for higher-order derivatives). Each derivative of e^-t is (−1)ⁿ e^-t.

When t approaches infinity, the function e^-t approaches zero, meaning the derivative -e^-t also approaches zero. When t is 0, the derivative of e^-t = -e^0, which is -1.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in t. Exponential Function: A function of the form e^x, where e is the base of the natural logarithm, representing exponential growth or decay. Chain Rule: A rule for differentiating composite functions. Quotient Rule: A method for finding the derivative of a quotient of two functions. Exponential Decay: A process where quantities decrease at a rate proportional to their current value, often modeled by functions like e^-t.